In this paper, it is discovered that in the differential geometry of arbitrary smooth plane curves there exists a solenoidal vector field * S**, i.e., the field with a property div

**= 0 (in a certain area**

*S***D*), having the following geometric meaning. Let {

*Lτ*} be a set of arbitrary smooth non-intersecting plane curves

*Lτ*with the Frene unit vectors

*,*

**τ**=**τ**(x, y)

*υ**=*(

**υ**(x, y)**is the unit tangent vector,**

*τ***is the unit normal of a curve**

*υ**Lτ*), and {

*Lυ*} be a set of orthogonal to them curves

*Lυ*with the Frene unit vectors

**and**

*υ**. The set {*

**η**= –**τ***Lτ*} fills by the continuous image some area

*D*and satisfies some general conditions. The vector field

**is expressed in terms of the Frene unit vectors**

*S****,**

*τ***of the curves**

*υ**Lτ*and equals the sum of the curvature vector

*d*of the curve

**τ**/ ds = (**τ**•∇)**τ**= k**υ***Lτ*∈ {

*Lτ*} and the curvature vector

*d*of the curve

**υ**/ dsυ = (**υ**•∇)**υ**= kυ**η**= –kυ**τ***Lυ*∈ {

*Lυ*}. Here

*s*and

*sυ*are natural parameters, i.e., the length of a curve being counted from its certain point along

*Lτ*and

*Lυ*, respectively,

*k = k(x, y)*and

*kυ = kυ(x, y)*are curvatures of the curves

*Lτ*and

*Lυ*, respectively. The symbol

*(*denotes a derivative of the vector

**a**•∇)**a****in the direction**

*a***. This property can be interpreted as existence in the differential geometry of plane curves of a conservation law (for the vector field**

*a***or for vector fields of unit vectors). A number of equivalent representations of the field**

*S****and equivalent forms of a conservation law div**

*S****= 0 is obtained.**

*S**